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A LIQUIDITY BASED ASSET PRICING MODEL
Bengt R. Holmstrom
Jean Tirole
June, 1998
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
A LIQUIDITY BASED ASSET PRICING MODEL
Bengt R. Holmstrom
Jean Tirole
No.
98-08
June, 1998
MASSACHUSETTS
INSTITUTE OF
TECHNOLOGY
MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
50
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
SEP
1 6 1998
LIBRARIES
LAPM: A
Liquidity-based Asset Pricing Model"
Bengt Holmstrom^
April
6,
Jean Tirole*
1998
Abstract
CAPM
The intertemporal
predicts that an asset's price is equal to the
expectation of the product of the asset's payoff and a representative consumer's intertemporal marginal rate of substitution. This paper develops
an alternative approach to asset pricing based on industrial and financial
corporations' desire to hoard liquidity to fulfill future cash needs.
Our
corporate finance approach to market finance suggests new determinants
of asset prices such as the distribution of wealth within the corporate
and between the corporate sector and the consumers. Also, leveradequacy requirements, and the composition of savings
affect the corporate demand for liquid assets and thereby interest rates.
The paper first sets up a general model of corporate demand
for liquid assets, and obtains an explicit formula for the associated liquidity premia.
It then derives some implications of corporate liquidity
demand for the equity premium puzzle, for the yield curve, and for the
state-contingent volatility of asset prices. Finally, the paper looks at some
macroeconomic implications of the theory. It shows that government may
be able to boost aggregate liquidity and enhance economic efficiency by
promoting job and asset price stability. On the liability side, long-term
deposits and equity investments, which depend on the consumers' endogesector
age
ratios, capital
nously determined liquidity needs, contribute to creating a feedback effect
between employment prospects and equity-like investments. On the asset side, orderly sales of real estate
by liquidity-squeezed
institutions
may
generate a Pareto improvement.
'The authors are grateful to Olivier Blanchard, Seven Busar, Mathias Dewatripont, DouDiamond, Peter Diamond, Arvind Krishnamurthy, Jean-Charles Rochet, Stephen Ross,
Jeremy Stein and Xavier Vives for helpful discussions. This research was supported by a
glas
grant from the National Science Foundation.
'MIT, Department of Economics
'IDEI and GREMAQ (CNRS UMR 5604), Toulouse,
MIT
CERAS (CNRS URA
2036),
and
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/liquiditybasedasOOholm
Introduction
1
(CAPM,
Starting with the capital asset pricing model
derived by Sharpe 1964,
Lintner 1965 and Mossin 1966), market finance has emphasized the role of
consumers' time preference and risk aversion in the determination of asset
The
prices.
intertemporal consumption-based asset pricing model
binstein 1976, Lucas 1978, Breeden 1979, Harrison-Kreps 1979,
Hansen-Jagannathan 1991) predicts that an
Cox
asset's current price
is
(e.g.,
et al.
Ru-
1985,
equal to the
expectation, conditioned on current information, of the product of the asset's
payoff and a representative consumer's intertemporal marginal rate of substitution (IMRS). While fundamental, this dominant paradigm for pricing assets
has some well-recognized shortcomings (see below), and there
is
clearly scope
and complementary approaches. This paper begins developing
for alternative
one such approach based on aggregate liquidity considerations. 1
Our
starting point
is
that the productive and financial spheres of the econ-
omy have autonomous demands
for
for financial assets
and that
their valuations
these assets are often disconnected from the representative consumer's. Cor-
porate
demand
order to
fulfill
for financial assets is
future cash needs.
driven by the desire to hoard liquidity in
In contrast to the logic of traditional asset
pricing models based on perfect markets, corporations are unable to raise funds
on the capital market up to the
level of their
expected income, and hence the
corporate sector will use financial assets as a cushion against liquidity shocks
(Holmstrbm-Tirole, 1996, 1998).
will
command
There
is
liquidity premia.
Financial assets that can serve as cushions
2
substantial evidence that firms
and banks hold
liquid assets as a
partial or full guarantee of future credit availability (see, e.g.,
Crane 1973 and
Harrington 1987). Companies protect themselves against future credit rationing
by holding securities and, especially, by securing credit lines and loan commitments with banks and other financial institutions. Lines of credit cover working
and back up commercial paper sales. Commitments provide longterm insurance through revolving credits, which often have an option that allows
the company to convert the credit into a term loan at maturity, and through
back-up facilities that protect a firm against the risk of being unable to roll over
its outstanding commercial paper. Companies pay a price for these insurance
services through upfront commitment fees and costly requirements to maintain
capital needs
'Liquidity in this paper does not refer to the ease with which assets can be resold. While
transaction costs (stamp duties, brokerage fees, bid-ask spread) have an important impact on
the pricing of individual assets, their implications for aggregate liquidity have not yet been
elucidated.
2
This theme
and other close
relates to Hicks' notion of "liquidity preference" for
substitutes.
He
monetary instruments
defines "reserve assets" as assets that are held to facilitate
in economic conditions and thus not only for their yield. For an
on the developments following Keynes (1930)'s and Hicks' contributions
to liquidity preference, see the entries by A. Cramp and C. Panico in the New Palgrave
Dictionary of Money and Finance.
adjustments to changes
historical perspective
compensatory balances.
Turning to the supply side, the provision of liquidity is a key activity of
the banking sector. Banks occur a nonnegligible credit risk, as the financial
condition of companies may deteriorate by the time they utilize their credit
facilities.
Furthermore, the rate of utilization of credit
substantially over time)
increase
when money
to meet demand.
events.
On
is
is
facilities
3
not independent across time.
tight, forcing
banks to scramble
(which varies
Credit use tends to
for liquidity in
order
Banks themselves purchase insurance against unfavorable
the asset side, they hoard low-yielding securities such as Treasury
notes and high-grade corporate securities.
On
the
liability side,
they issue long-
term securities to avoid relying too much on short-term retail deposits. Within
the banking sector, liquidity needs are managed through extensive interbank
funds markets.
A
corporate finance approach to asset pricing has several potential bene-
it enlarges the set of determinants of asset prices.
In contrast to
consumption-based asset pricing models, in which the net supply of financial
assets is irrelevant for asset prices because they are determined exclusively by
real variables, our model features a feedback effect from the supply of assets
to the real allocations. The corporate sector is not a veil and the distribution
of wealth within the corporate sector, and between the corporate sector and
the consumer sector matters; in particular, the capital adequacy requirements
imposed on banks, insurance companies and securities firms, and the current
First,
fits.
and nonfinancial institutions affect corporate demand for
and thereby asset prices. Second, government influences the
aggregate amount of liquidity through interventions such as open market operations, discounting, prudential rules and deposit insurance (see Stein 1996) and
this impacts liquidity premia and asset prices. Consumption-based asset pricing
leverage of financial
financial assets
models, provided they exhibit a form of Ricardian equivalence, have little to say
about the impact of such policies.
Because the consumption-based asset pricing model is entirely driven by
consumers' intertemporal marginal rates of substitution (IMRS), it has several
empirical limitations. 4 First, it tends to underpredict both equity premia (see,
5
e.g., Mehra-Prescott 1985) and Treasury bill discounts.
Second, changes in
Calomiris (1989) argues on the basis of a survey of US market participants that the Central
is sometimes forced to inject market liquidity during credit crises because of bank loan
Bank
commitments.
We view these
complement it with
5
facts less as critiques of the theory
As Aiyagari-Gertler (1991)
note, "reasonably parametrized versions [of the intertemporal
asset pricing model] tend to predict too low a risk
and do not account
than as indicators of the need to
liquidity considerations.
premium and too high a
risk-free rate,"
annual real return on stocks and 1
percent secular average return on Treasury bills. A number of papers have used nonseparable
preferences to address the equity premium puzzle (with mixed success; see Ferson 1995 and
Shiller 1989 for reviews), while the other puzzle — the low risk-free rate — has been largely
ignored as pointed out by Weil (1989) and Aiyagari-Gertler.
for the 7 percent secular average
marginal rates of substitution (discount rates) induce asset prices to covary.
Shiller (1989, p. 346-8) notes, "prices of other speculative assets, such
as bonds, land, or housing, do not show movements that correspond very much
But as
at all to
movements
in stock prices."
Third, the consumption-based asset pricing model does not properly account
for
to
the facts that the yield curve
medium term
is
on average upward sloping
bonds), that government intervention affects
that long-term bonds feature substantial price volatility (see,
often argued that the term
premium
up
and
(at least
its
slope,
e.g., Shiller
1989,
from the price
risk of long-term bonds. However, this argument is not supported by a complete
market model like CAPM, because price risk per se only entails a reshuffling
of wealth among investors and involves no aggregate risk that would deliver a
chapter 12.)
It is
results
premium.
Fourth, the consumption-based asset pricing model has not guided the im-
portant advances
ARCH
in
Autoregressive Conditional Heteroskedasticity
(ARCH)
models. 6
models and their generalizations allow the covariance matrix of inno-
vations to be state-contingent in order to reflect observations by Mandelbrot
Fama (1965) and Black (1976) that variances and covariances of asset
change through time and that volatility tends to be clustered across time
(1963),
prices
and across
A
assets.
proper treatment of these empirical discrepancies of the consumption-
based asset pricing model lies outside the limited scope of this exploratory paper.
Let us briefly indicate though how the liquidity approach could help to bridge
the gap between theory and empirics.
First, concerning the equity premium puzzle we find that Treasuries and
other high-grade securities offer better insurance against shortfalls in corporate
earnings and other liquidity needs than do stocks.
To
highlight this point,
we
assume that consumers are risk neutral, so that stocks and bonds
would trade at par in the standard model; yet in our model bonds command
a liquidity premium and trade at a discount relative to stocks.
Second, by assuming risk-neutral consumers, we eliminate consumer IMRS's
as drivers of asset price movements. Instead prices will be responding to changes
in corporate IMRS's, which in turn are influenced by corporate demand for
For instance, unlike
liquidity.
The theoretical implications are different.
the consumption CAPM, our model implies that an increase in the supply of
In
liquidity drives bond prices down at the same time as stock prices go up.
empirical work, we also believe that the corporate sector may offer a better
measure of changes in the marginal purchaser's IMRS than do corresponding
attempts to study partial participation among consumers. (For a recent effort,
deliberately
alike
see Vissing-Jorgensen, 1997).
6 See, e.g., BoUerslev
et
al. (1992), Engle et al. (1990) and Ghysels-Harvey-Renault (1996).
methods, though, have been used on an ad hoc basis (for example by allowing the
betas in a CAPM model to change over time) in order to improve the fit of the intertemporal
asset pricing model.
ARCH
Third, in our model the yield curve
maturities as liquidity buffers,
assets
and by the anticipated
and
is
is
determined by the value of various
by the availability of other
also affected
institutional liquidity needs. This generates richer
patterns for the yield curve than in a consumption-based theory.
is state contingent and exhibits serial
Under some conditions, the price volatility of fixed-income securities
Fourth, in our model price volatility
correlation.
covaries negatively with the price level (as Black 1976 notes, volatilities, which
covary across assets, go up when stock prices go down.)
income
securities
embody an
option-like liquidity service.
probability of a liquidity shortage, the option
will
be sensitive to news about the
shortage
is
future.
is
"in
When
Intuitively, fixed-
When
there
the money" and
is
a high
its
price
the probability of liquidity
sufficiently low, the price of the option goes to zero
and
will
not
respond much to news.
The paper
is
organized as follows. Section 2 illustrates the determination of
premia through a simple example. Section 3 sets up a general model of
corporate liquidity demand. This model shows how departures from the ArrowDebreu paradigm generates liquidity premia. Sections 4 and 5 demonstrate
that the liquidity approach delivers interesting insights for the volatility of asset
prices and for the yield curve.
The final section 6 studies two examples with
endogenous date-1 asset prices to illustrate the possibility of multiple equilibria
and the role that policy measures can play in ensuring that the better equilibrium gets selected. The first example (section 6.1) suggests that unemployment
insurance can provide liquidity that supports long-term investments. The second example (section 6.2) illustrates the problem with real assets as sources of
liquidity: such assets have a low value precisely when the aggregate need for
liquidity
liquidity
2
is
high.
Liquidity premium: an example
There are three periods,
=
and a continuum (of mass 1) of
one project. Entrepreneurs are risk neutral
and do not discount the future. They have no endowment at date and must
turn to investors in order to defray the fixed date-0 set up cost, /, of their project.
At date 1, the project generates a random verifiable income x. The realization
of x is the same for all entrepreneurs and so there is aggregate uncertainty. The
distribution G(x) of x is continuous on [0,oo), with density g(x) and a mean
greater than /.
At date 1, the firm reinvests a monetary amount y > 0, which at date 2
t
0,1,2, one good,
identical entrepreneurs, each with
generates a private benefit by
—
2
^
f° r
the entrepreneur, with b
private benefit cannot be recouped by the investors;
The only noncorporate
its
financial asset
is
it is
>
1.
This
nonpledgeable.
a Treasury bond. 7
At date
the
The only feature of a Treasury bond that is going to be relevant for our analysis is that
payoff is exogenous. It could equally well be the output of some similarly exogenous asset
government issues
in
every state.
L bonds that mature at
The
date-0 price of a bond
and
one unit of the good
of the Treasury
date
1
is q.
The repayment
yield
bond is financed through taxes on consumers.
Consumers (investors) are also risk neutral and do not discount the future.
They value consumption stream (co,Ci,C2) at cq + Cj + c^. A key assumption
is that consumers cannot individually commit to provide entrepreneurs with
funds at date 1, because they cannot borrow against their future income. 8 The
consumers' preferences imply that they will hold only assets whose expected
rate of return is nonnegative. This implies in particular that q > 1 If q > 1,
consumers hold no Treasury bonds and we will refer to q — 1 as the liquidity
premium. The liquidity premium can be strictly positive, because consumers
cannot commit to date-1 payments and hence cannot short-sell assets.
A contract between a representative entrepreneur and the investors specifies
a quantity L of Treasury bonds to be held by the firm and an income contingent
reinvestment policy y(x). Because consumers cannot commit to provide more
funds at date 1, the reinvestment policy must satisfy
.
y{x)
<x+
L.
(1)
For notational simplicity, we will assume that investors receive
easy to show that this
is
indeed the case
if
x + L — y.
the expected date-1 income
is
It is
not
too large. 9 Investors expect a nonnegative rate of return, and so
E [x-I-y(x)-(q-l)L}>0,
(2)
a date-0 expectation, and expectations are taken with respect to the
A competitive capital market guarantees that (2) is satisfied
with equality. Because investors break even, entrepreneurs receive the social
where
is
random
variable x.
surplus associated with their activity. Figure
Figure
The optimal
1
1
summarizes the timing.
here.
contract between a representative entrepreneur and the in-
vestors maximizes the social surplus generated by the entrepreneurial activity,
like land.
8 One can rationalize this assumption by assuming that individuals can credibly claim that
The tax authority in contrast has information about individual
they have no date-1 income.
An alternative interpretation is that unborn generations of consumers cannot make
See our 1998 paper for a
commitments, but the government can do so on their behalf.
more extensive discussion of the theoretical foundations and for the empirical validity of these
earnings.
assumptions.
9
When expected income is very large, that
themselves, the results below apply with fi = 1.
is,
when
firms generate substantial liquidity
21
-
Eo by(x)
(</(*))
subject to constraints (1) and (2). Letting /z
constraint (2), this optimization amounts to
>
1
denote the shadow price of
2
max iEo
(y(*))
\by(x)
+ n[x - I - y(x) -( 9 -i)l]J
subject to the liquidity constraint
y(x)
<x + L
for all x.
For fixed L, the solution of the unconstrained program
y*
= b- n,
and the solution to the constrained program
y(x)
=
is
min(y*,x
(3)
is
+ L),
(4)
so firms are liquidity constrained in low-income states.
Let us turn to the date-0 choice of
zation,
L
is
liquidity.
From our previous
characteri-
chosen to maximize
g{x)dx
.
Assuming L >
(y')
-
+ y* + (q-
n[I
the first-order condition
\)L
- x]
g(x)dx.
is
rv'-L
-"jf [^f^-
1
g(x)dx.
(5)
Let
6"
(I+L)
m(x)
-
1
for
for
x < y* - L
x > y* - L.
Then
q- =
1
Eo[m(x)].
(6)
The
liquidity
premium
equal to the expected marginal value of the liquidity
(a; < y* — L), an extra unit of liquidity
is
service. In states of liquidity-shortage
its reinvestment by 1 and the private benefit by
This marginal private benefit, expressed in monetary terms, is
equal to [b — {x + L)]/fi. The increase in reinvestment has monetary cost 1.
This yields the expression for the liquidity premium (6).
We have looked for an equilibrium {<7,/f} in which Treasury bonds command
a liquidity premium. Such an equilibrium satisfies the break-even constraint (2)
allows the firm to increase
b
—
(x
+
L).
(with equality) and the asset pricing equation (5), for L = L (because q > 1,
do not hold Treasury bonds), and for the optimal reinvestment policy
investors
y(-) defined
by
(3)
and
(4).
is free,
firms
It is
L<
easy to show that
L*\ and
If
the solution to this system yields q
that is, even
command no liquidity premium (q = 1);
may not hold all available liquid claims.
Treasury bonds
i)
>
there exists an L* such that q
1
if
<
if
1,
then
liquidity
and only
if
the value of the marginal liquidity service m(-) and the price of
liquid claims q are monotonically decreasing in L. These results are illustrated
in figure 2, where Tnr,(-) denotes the marginal liquidity service for bond supply
L.
ii)
When L =
liquidity
L3 > L* the economy has surplus liquidity, and there is no
premium. 10 Hence, bond prices are low. The cases L = L\ or L<i
,
depict the interesting case of scarce liquidity.
Figure 2 here.
Remark: Note that the value of
liquidity
m(x)
is
linear in
x
in this
Linearity requires that the date-2 payoff be nonverifiable, that
is,
example.
the payoff
is
purely a private benefit. Alternatively, the date-2 payoff could be verifiable but
not fully pledgeable as in Holmstrom-Tirole (1998), for instance, because moral
hazard problems require the entrepreneur to keep a stake in the final payoff.
Under mild conditions, the value of liquidity m{x) would then differ from that
depicted in figure 2 only in that the decreasing part would be convex rather
than linear. 11
LAPM
3
Let us
now develop a more
0, 1,2.
At date
10 To endogenize
1,
general framework.
a state of nature
L we would have to
w
There are three periods, t —
economic agents.
in fi is revealed to all
introduce a cost of issuing Treasury securities, such
when debt is retired. See our 1998 paper for more
as a deadweight loss of taxing consumers
detail.
'^The assumptions are: The reinvestment y produces date-2 income /(y) with probability p
and with probability 1-p. The entrepreneur can work (p = pn) or shirk (p = pi, = P//- Ap).
Shirking generates a private benefit Bf(y). Letting po = p//(l - (S/Ap)] we have m"(x) >
if3 P0 (/") 2
+ (l-M/')/'">0.
There
is a further resolution of uncertainty at date 2, but in our risk neutral
framework, only date-1 expectations matter so we need not specify the date2 random events. The state of nature u) may include the date-1 profits of
the various industries in the corporate sector (as in the example above), their
date-1 reinvestment needs (as in our 1998 paper ), news about the prudential
requirements or government policy, or signal/prospects about date-2 revenues. 12
As
• Investors.
example, and in order to highlight the departure from
in the
the canonical asset pricing model,
we assume that
have an exogenously given discount
rate,
investors are risk neutral
normalized at zero. That
value consumption stream (co,Ci,C2) at cq
+ C\ +
c<i.
is,
and
investors
One could assume more
generally that investors have endogenously determined
and possibly stochas-
Similarly, the implicit assumption that investors face
no
be relaxed (see section 6.1).
• Noncorporate claims. At date 0, there are
noncorporate assets, k =
l,...,K such as Treasury securities or real estate. The return on asset k at date
discount factors.
tic
liquidity needs could
K
that
1,
is,
the date-1 dividend plus the date-1 price,
The mean
0.
where
Et[-]
return on each asset
is
is
equal to 0%
normalized to be one:
=
0fc(u>)
>
=
1,
Eo[9k[y>)\
denotes the expectation of a variable conditional on the information
available at date
t.
Let Lk denote the supply of asset k. At date 0, asset k trades
where qk > 1 from the nature of consumer preferences. The
at price qk per unit,
premium on asset k is equal to qk — 1.
Note that the returns {9k(u>)} are exogenously given. In section 6 we will
analyze two examples with endogenous returns.
Note also that claims k =
do not include claims on the corporate sector (shares, bonds, deposits,
1,
liquidity
.
.
,
.
K
CDs,...).
We will
later provide valuation formulae for these.
Our model
• Corporate sector.
treats the productive
and
financial sectors
The Appendix
The corporate sector
as a single, aggregated entity, called the "corporate sector."
provides sufficient conditions validating this approach.
invests at dates
and
1
and
receives proceeds at dates
1
and
2.
Let I denote
the corporate sector's date-0 gross investment (or vector of gross investments)
in productive (illiquid) assets.
Its date-0 net investment, N(I), is equal to
the difference between the gross investment and the productive sector's capital
contribution at date
had no
The
initial
(in
the example above, N(I)
=
I since the entrepreneurs
wealth).
net investment
N(I)
is
only part of the investors' date-0 contribution to
The corporate sector also purchases noncorporate
a t date 0. The investors' date-0 outlay is thus
the corporate sector.
{Lk}k=i,...,K
N(I)
In equilibrium all claims
+ J2 k qk L k
commanding a
held by the corporate sector (Lk
=
assets
.
liquidity
premium
(qk
>
1)
must be
Lk), because consumers are risk neutral.
For a more general representation of consumer preferences, could also include shocks to
investors' rate of time preference.
At date
1
D(u>, L(uj))
,
state w.
,
the corporate sector selects a decision d
where
The
L(u>)
is
decision vector d includes
all real
d(u>)
from a feasible set
decisions within the corporate
and production decisions
sector such as reinvestments
itly, it
—
the net liquidity available to the corporate sector in
among
for
every firm.
Implic-
support
financial contracting between
also includes reallocations of liquidity
firms, required to
This presumes fully efficient
For instance, scarce liquidity
these real decisions.
may be allocated
Holmstrom-Tirole 1998 for
more detail). If the corporate sector does not pay insiders (entrepreneurs) anything at date 1 (see below), then
firms in the corporate sector.
efficiently at
date
1
by
financial intermediaries (see
Recall that investors cannot commit to bringing in new funds at date
the amount that is held in liquid assets L(u>).
Assumption
For
all u),
1
1
beyond
(opportunity-enhancing liquidity)
L\ and L2:
if
L\
<
Li-,
then D(lj,L^)
C
D(w,Z,2).
In general, an increase in liquid reserves strictly enlarges the set of feasible
corporate policies when financial markets are imperfect.
earlier
example,
ui
= x,
and
D(u>, L)
—
y
For instance, in our
< x + L}.
{y
For a given state u, let R(I,u,d) denote the total expected intertemporal
(date 1 plus date 2) payoff from illiquid corporate assets (in the example, R =
x+\by —
\
\\— y)- R includes pledgeable and nonpledgeable returns on
illiquid
assets, but excludes the return ^2 k 8k(u)L k on noncorporate securities. Let
r(I,u,d) denote the corresponding pledgeable income from illiquid assets, that
is the expected intertemporal income that can be returned to investors (in the
date
0,
=
— y).
Accounting for noncorporate assets purchased at
the corporate sector can return at most
example, r(I,oj,
d)
x
r{I,u),d(u}))
+ ^Ok {w)L k
k
to investors. Let
B(I, v, d(w))
=
R(I,u>, d{u>))
-
r{I,u, d(w))
The fact that B > is critical for
demand for liquidity in our model.
Let t(u>) denote the amount of pledgeable income that is paid out (at date 1)
corporate insiders (entrepreneurs) in state u>. The net liquidity available for
denote the nonpledgeable portion of income.
generating a corporate
to
reinvestment in state
The corporate
u> is
then ^2 k 6 k (Lo)L k
sector solves:
10
—
t(u>).
max
{Eo[R(I,u, d(w))]
-
/
-
V
(g fc
-
l)L k \
,
subject to the investors' break-even condition:
E
[r{I,u,d(v))
-
and to date-1 decisions being
d(w) G
Assuming that the
t(u)]
+ J2 k Lk ^ N(I)+^2 k qk Lk
(7)
,
feasible:
D
(w,
J2 k Bk{v)L k - t(w))
investors' break-even constraint
is
(8)
.
binding, this program can
be rewritten as
max
{/
{Eo [B(J,w,dM)
+ t(w)] +N(I) - 1}
)}
subject to (7) and (8).
Let n > 1 denote the shadow price of the break-even constraint in this latter
program, and define
d
\-
I
(B(I,u,d)+nr(I,cj,d)
max
<
(9)
d£D(u>,L) [
/i
L=L(u>)
as the marginal liquidity service (expressed in terms of pledgeable income) in
w assuming
state
tion
1
that the available liquidity
implies that
m(w) >
Assume that there
is
L(u>)
=
^2 k 6 k (u))L k
Assump-
.
for all w.
one state of nature
exists at least
in
which there
is
excess
is in the interior of the feasible
which the
is a mild assumption and is satisfied in all our examples.
implies (see the Appendix for more detail) that pledgeable income is never
that
liquidity,
is,
decision d(ui)
in
decision set D. This
It
redistributed to the corporate sector in states of liquidity shortage
if
m(o>)
>
0),
and so the available
liquidity in equation (9)
is
(t(oj)
—
appropriately
defined. 13
Optimization with respect to
qk
-
1
Lk
=
,
at equilibrium
Eo[O k (u)m{u)l
market
prices,
yields
14
(10)
or, equivalently
qk
= Eo[ek {cj)[l+m(u})]].
13 By
the same reasoning, we will be able to ignore state-contingent liquidity withdrawals
the other programs in the paper.
14 Note that we
assume that the corporate sector as a whole takes prices as given. Price
taking presumes that there is competition for assets within the corporate sector. In section
6.2 we consider cartel behavior.
£(•) in
11
Like risk premia, liquidity premia are determined by a covariance formula, but
time involving the intertemproal marginal rate of substitution 1 + m(w)
this
of the corporate sector.
income
An
asset's liquidity
premium
is
high when
it
delivers
which liquidity has a high value for the corporate sector.
For completeness, we can finally introduce external claims on the corporate
sector (shares, bonds, etc.).
Let Lj be the date-0 supply of claim j paying
Oj(u) at date 1 in state of nature u. The set of external claims, j = 1,
J,
in states in
.
must
.
.
,
satisfy
J^ Oj^Lj
The
= r{I,u,d(u>)) +
^
date-0 prices of such claims, {qj} =1
e k {w)L k
-
t(w).
fc
,
,
will
be given by
Single state of liquidity shortage.
Suppose liquidity is scarce in a single state, u>#, which has probability /#.
According to (10), the liquidity premium on asset k is then proportional to the
asset's payoff conditional on the occurrence of the bad state:
qk
-
1
=
jH0k{u H )m H
(11)
.
This linear relationship yields the following
Qk-l
qe-l
__
OkjUH)
,
OeM
K
2)
'
which implies a one-factor model of liquidity premia. We can take the bond
That is, if g& is the date-0 price of a bond delivering
one unit of good at date 1, the liquidity premium on asset fc perfectly covaries
price as the single factor.
with the liquidity premium on the bond: 15
qk-
1
=6 k {u} H ){qb -
1).
With more than one state
of scarce liquidity, (10) results in a multi- factor model
where the factors can be chosen as the liquidity premia of any subset of assets
that spans the states in which liquidity shortages occur.
The Arrow-Debreu economy.
Assets do not command a
liquidity
premium under the
following additional
assumptions:
This assumes that the asset's payoff 6k does not vary conditional on
then 9 k {u>n) should be replaced by E{f>k i^h) in tne formula below.
\
12
a;//.
If it varies,
(fully pledgeable income): For all /, u, d, R(I, w, d) = r(I, u, d).
Assumption 3 corresponds to the absence of agency costs and private (non-
Assumption 3
pledgeable) benefits.
Assumption 4
(efficient contracting): For
max
r(I,u),d)
=
d£D(u,L)
all
max
w and
L,
all
r(I,u>,d)
d€D(w,0)
According to this assumption, liquidity is valueless when it comes to maximizing pledgeable income. To understand its role, suppose that the corporate
sector hoards no liquidity, and so L = 0. Suppose that in state of nature u>,
there exists
L
such that
r(I,w,d*(<J,L))>r{I,w,d*(cj,0)),
where d*(u,L) denotes the maximizing pledgeable income decision in state w
given liquidity L. The corporate sector could at date 1 borrow L and pledge
r(I,u),d*(tJ, L)) — L + L > r(I, w,d*(w,0)), a contradiction.
for all u, since
Given Assumptions 3 and 4, m(u)) is equal to
\
max
=
max
B(I,uJ ,d)+nr(I,uJ ,d)
\
{r(I,uj,d)\
d£D(w,L(w))
=
max
deD(w.o)
Thus,
all liquidity
{r(I,
u),d)\
l
v
'
premia are zero
in
.
an Arrow-Debreu economy.
Information filtering and volatility
4
in the introduction, many recent advances in empirical finance
were motivated by the observation that conditional variances and covariances
change over time. It is well-known, for instance, that volatility is clustered,
As we noted
that asset volatilities (stock volatilities,
and that stock
bond
volatilities across maturities)
move
with bad news. 16
This section does
not attempt to provide a general theory of the impact of liquidity premia on
volatility. Its only goal is to suggest that a liquidity-based asset pricing model
has the potential to deliver interesting insights into state-contingent volatilities.
together,
volatility increases
16
Black (1976) attributes the last fact to the "leverage effect" (equity, which is a residual,
ratio increases.) However, this leverage effect does not seem
to account for clustering of volatility and comovements between stock and bond volatilities.
moves more when the debt-equity
13
Example
4.1
Let us
first
return to the example of section
2.
In this
example with nonverifiable
second-period income, the liquidity benefit of the Treasury bond
is a put option,
income x until it hits zero. We
also observed that with verifiable second-period income and under some mild
regularity conditions, m(x) decreases and is convex until it hits zero.
Suppose now that news arrives intermittently between dates
and 1 containing information about the realization of x at date 1. Specifically, suppose
that there are TV news dates between
and 1 (the first distinct from date and
the Nth equal to date 1). Assume further that the realization of x is given by
either an additive or a multiplicative process
since
m(x) decreases
linearly with first-period
= Xq+^2 Vm
771
(13a)
=1
N
m=n+l
=
=
x
where the increments
The early
r\
m
x [x" =1 Vm]
(13b)
x n [x^ =n+1 r]m ]
for all n,
are independently distributed.
accrual of information about the state
will
lj
have no impact on the
and hence no retrading of financial contracts occurs
between dates and 1. Yet, we can price Treasury bonds by arbitrage at each
subdate n. Contingent on the available information at subdate n, summarized
by x n
optimal decision rule
d(-)
,
qn (x n )
-
1
= En [m(x)
\
xn ],
(14)
where Z5n [-|-] denotes the conditional expectation given information at subdate
(Formula (14) is derived formally for the general framework in section 4.2.)
It can be shown that, provided m' <
and m" > 0, for either the additive
n.
(13a) or the multiplicative process (13b),
—
[En [[qn+ i(x n+ i) -
we have 17
qn (x n )}
2
\
x n }] <
0.
(15)
17 The proof for the
multiplicative process follows from that for the additive process by
taking logs in (13b). Indeed if m(') is decreasing and convex, m(e x ) is also decreasing and
convex
in x,
where x =
logo:.
14
bond
In words, the volatility of the Treasury
price
state contingent,
is
and the
The simple logic
the money and low when it is out
higher the volatility, the worse the prospects for the economy.
is
that volatility
high
is
when the option
is in
of the money.
Remark:
example with nonverifiable second-period income, the Blackbond price if
a continuous time geometric Brownian motion.
In the
Sholes formula yields an explicit formula for the volatility of the
the process
4.2
is
General
LAPM
Let us investigate more generally the impact of news about the state of nature
in
the
LAPM
are subdates
framework of section 3. As
= 1,...,./V between dates
in
n
accrue about the date-1 state of nature
the example, we assume that there
and
1,
at which informative signals
Thus, the market's information about
the state of nature at date 1 gets refined over time. Let a n denote the market's
information at subdate n with o> = w. We let E[- cr n ]denote the subdate-n
expectation of a variable conditional on the information available at time n.
The corporate sector purchases quantity L k of liquid asset k at date 0, and can
u>.
|
afterwards reconfigure
quantity
Lfc(cr n )
its
portfolio so that
at subdate n. Asset
holds (information contingent)
it
equilibrium price given information
A;'s
an
denoted qk{on )Consider the problem of maximizing the corporate sector's expected payoff
subject to the investors' date-0 break-even condition and date-1 decisions being
is
feasible:
18
{E
max
[B(I,u, d(u))\
+ N(I)
1}
subject to
E
[r(I,w,d(cj))]
>
N(I)
+ ^qk L k
N
+E 53 X^ qk
.
—Eq
k
n=l
(16)
^ [M
£)Mw)L
fc
a") - L k
(
ff n-l)]
(w)
L k
and
The maximand of
this
program
is
the same as
constraint.
15
in (3)
upon substitution of the budget
,
.
d(w)€D(w,]£0 (w)Ma;))-
(17)
fc
A
few comments are in order. First, a
so the set of feasible decisions
is
measurable with respect to
is
uj,
and
indeed a well-defined function of the state
Second, the date-0 contract with investors specifies some portfolio
adjustment at each date. We ignore the possibility that contemplated portfolio adjustments may require a net contribution by investors at subdate n
— Lk (crn _i)] > 0). While such a contribution could occur if
(5Zfc Qkipn) [L k (<Jn )
the portfolio adjustment raised the investors' wealth conditional on a n it would
not occur if the adjustment reduced it, since the investors would be unwilling
ex post to bring in new funds, and they cannot ex ante commit to do so. However, if in equilibrium qk > 1, then L-K (o n ) = L k for all a n is an optimal policy,
and so investors do not have to contribute at intermediate dates. The fictitious
subdate-n reshuffling of liquid assets between the corporate sector and the rest
of the economy is, as in Lucas (1978), only used to price financial assets at an
of nature.
,
intermediate date.
As before, we let
be the multiplier of the break- even constraint in (16)
and define the marginal liquidity service m(uS) as in (9). Taking first-order
conditions we find that for each liquid asset k £ {1, ..., K} and for each subdate
nG{l,...,7V-l}:
/j,
qk
qk
= 1 + Eo [0k (w) m (w)]
= E [qk{<Tn
)\
and
9fcK)
= En
[0 k
(u)
[1
+ m(w)]
|
a„]
(18)
Asset prices (or liquidity premia) form a martingale because there is no liquidity
service within the periods where news arrives. Only at the last subdate (date
1) will
the liquidity premium disappear and hence the martingale property fail.
reflects the fact that firms are indifferent
Note that the martingale condition
regarding the timing of the purchase of liquidity, as long as the purchase is made
before the final date. The martingale condition stems from arbitrage within the
budget constraint.
Define a "generalized fixed-income
unchanged as news accrues, that is,
investors'
En [Ok (u)\an =
]
security," as
l,
For simplicity, we focus on these securities
then write (18) as
16
for all
in
one with expected return
n and a n
(19)
.
the rest of this section.
We
can
,
qk{o n )
-l = En
[Ok (w.)
m(uj)
an
|
(20)
]
Condition (19) rules out volatility stemming from news about the assets' dividends. Such volatility must be added (with a correction depending on the
covariance with the innovations about liquidity needs) to our price formulae
when expected
Asset
payoffs change over time.
a single state of liquidity shortage.
volatility in the case of
Assuming that there is a single state (state u>h) of liquidity shortage, let ///(<?„)
denote the posterior probability of the bad state of nature at subdate n, conditional on the information available at that subdate. Let
3(<7„)
///(Cn+l) -/tf(On)
= En
l2
///(<7n)
denote the relative variance of the posterior probability. 9(er7V )
the informativeness of the signal accruing at subdate n + 1.
is
a measure of
Note, from (20), that the ratio formula (12) continues to apply with qk(c n )
in
place of qk- So at subdate
n and
-
1
qtipn)-
i
gfc(gn)
for
_
~~
any two assets k and
Ok (uh)
Ot(u H )
I,
.
~
k
(12')
'
Under the (strong) assumption of just one liquidity constrained state, and with
a government bond on the market, all assets with constant expected dividend
are priced according to a linear formula involving the liquidity premium on that
bond. This is a close analog to the CAPM.
Contingent
volatilities
and
clustering.
Let Vk and Vk denote the absolute and relative
volatilities of
the price of
asset k conditional on information a„:
Vk {a„) = En
\[qk {an+ i)
-
qk{on )\
(21)
and
Vk(?n)
gfc(On+l) ~<?A:(On)
= En
<7„
(22)
<Ik{On)
With only one liquidity-constrained state, the price in information state an of
an asset k with constant expected dividend is, as we have seen,
<7fc(tfn)
-
1
=
fH{On)0k (uh)
17
%•
(23)
This immediately yields our main formulae:
Vk {an = 9(<7B
)
)
Vk (an = 9(a.)
2
{an )
-
l]
9fc(CT„)
-
1
[qk
(24)
,
(25)
)
(26)
Ufc(0n)=A
Equations (24) and (25) state that,
in
MU
€ (<T n ),
(27)
absolute as well as relative terms, the
an asset's price is proportional to the square of the asset's liquidity
premium. Because prices move slowly, an immediate corollary is that an asset's
price volatility is serially correlated; that is, we predict temporal clustering of
an asset's volatility.
Equations (26) and (27) state that the volatility ratio of two assets is constant
over time. Assets volatilities move together because they are driven by the same
news concerning the likelihood of a liquidity shortage.
Returning to formulae (24) and (25), we note that volatility is also proportional to the informativeness S(cr n ) of the signal accruing at subdate n+1. With
two states, this informativeness measure is generally state-dependent, rendering
these results less useful. To illustrate a case with constant informativeness, suppose that information accrues according to a Bernoulli process: At each subdate
n, with probability A G (0, 1), the market learns that the economy will not be in
the bad state of nature (///(<7n ) = 0). If that happens, the economy becomes an
"Arrow-Debreu economy," in which assets command no liquidity premium and
hence qk{&m) = 1 for all m>n. In this absorbing state, informativeness may
be taken equal to 0. With probability 1 — A, the economy remains an "LAPM
~
economy" and ///(<7 n ) = (1 — \) N n A simple computation shows that in this
volatility of
.
example:
If
There is price volatility as long as the liquidity premium is strictly
news accrues that the economy will be replete with liquidity, the
premium goes
to zero as will the volatility of prices for
expected payoffs.
18
all
positive.
liquidity
assets with constant
The
5
yield curve
5.1
The
The
theoretical
slope of the yield curve and price risk
and econometric research on the term structure of interest
rates
traditionally views the corporate sector as a veil in the sense that the yield curve
is
management (ALM). Many
not influenced by asset liability
that corporate liquidity
demand
affects the
term structure.
believe, however,
First, while
debt
markets are segmented, there is enough substitutability across maturities to induce long and short rates to move up and down together (Culbertson, 1957).
Duration analysis, stripping activities and more generally financial engineering and innovation (answering the question of "who is the natural investor for
the new security?") provide indirect evidence for segmentation. A number of
factors such as fiscal incentives, the creation of pension funds, new accounting
and prudential
cial sectors
rules for intermediaries,
are likely to affect the
influencing the term structure.
and the leverage of the
demand
real
and
for maturities differentially,
finan-
thereby
Second, the maturity structure of government
debt seems to play a role in the determination of the term structure, a fact that
19
is not accounted for in Ricardian consumption-based asset pricing models.
is most commonly upward sloping, although it may occabe hump-shaped or inverted or even have an inverted-hump-shape (see,
Campbell et al., 1996, Campbell, 1995, and Stigum, 1990). The substantially
higher yield on 6 month- than on 1 month-T bills has been labelled a "term
premium puzzle." 20 It is often argued that an upward-sloping yield curve re-
The
yield curve
sionally
the riskiness of longer maturities. Investors, so the story goes, demand
a discount as compensation for price risk (which presumably is correlated with
consumption if the standard model applies). This argument is based on an
flects
CAPM.
would be worthwhile, though, to provide a precise defCAPM is about the coupon risk of assets.
Coupon risk relates to uncertainty about dividends, or, more generally (to encompass uncertainty about preferences and endowments), to uncertainty about
analogy with
It
inition of the notion of "price risk."
the marginal utility of dividends.
Price risk may stem from coupon risk, but it need not. Consider an intertemporal Arrow-Debreu endowment economy (as in Lucas 1978). In this
economy, early release of information about future endowments is irrelevant in
that it affects neither the real allocation nor the date-0 price of claims on future
endowments. On the other hand, release of information affects asset prices,
inducing price risk. In an Arrow-Debreu economy, the date-0 price of claims
on date-2 endowments can be entirely unrelated to the variance of their date-1
For example, the Clinton administration has begun to shorten the average maturity of
government debt to take advantage of lower short term yields.
There are a number of other stylized facts: short yields move more than long yields; longterm bonds are highly volatile; and high yield spreads tend to precede decreases in long rates.
Also the yield curve tends to be natter when money is tight.
19
,
price (or to the covariance of price
Price risk per se
is
and some measure of aggregate uncertainty).
not an aggregate risk and thus need not affect asset prices.
Returning to the yield curve, Treasury bonds are basically default-free. Uncoupon risk (for nominal
bonds), which in turn affects prices. Inflation uncertainty clearly plays an important part in explaining the price risk of long-term bonds. But for short-term
bonds the connection is less obvious. A bond that matures in less than a year is
quite insensitive to inflation, at least directly. Indirectly, swings in the price of
long-term bonds will of course influence short-term prices as long as maturities
certainty about the rate of inflation, however, creates a
are partially substitutable.
explain the term
premium
Even
so, inflation-induced price risk
can hardly
puzzle.
Our point is to caution against drawing hasty conclusions about the link
between price risk and the slope of the vield curve. A theoretical justification
based on the standard CAPM logic cannot be provided, because in a complete
market, price risk stemming from early information release will not carry any
risk premium.
This opens the door for alternative theoretical approaches to
analyzing the yield curve.
Our liquidity-based asset pricing model offers one
possibility.
5.2
Long-term bonds and the Hirshleifer
effect
In order to obtain some preliminary insights into the effects of liquidity on the
Assume
term structure, let us again return to the example of section 2. 21
that the government at date
issues two types of bonds: I short-term bonds
yielding one unit of the good at date 1, and L (zero coupon) long-term bonds
yielding 6 units of the good at date 2. We allow for a coupon risk on longterm bonds, so let 6 be a random variable with support [0,oo), density h(6),
cumulative distribution H(0), and mean Eq(6) = 1. As discussed above, 8 can
be interpreted as the date-2 price of money in terms of the good. The case of
a deterministic inflation rate (which can be normalized to 0) corresponds to a
= 1. We let q and Q denote the date-0 prices of
spike in the distribution at
short- and long-term bonds. A long-term price premium corresponds to q > Q.
Treasury bonds are the only liquid assets in the economy
In the example the date-1 income x is assumed to be perfectly correlated
across firms. Let g(x) and G(x) denote the density and the cumulative distribution of income x. Assume that 6 and x are independent. In this economy,
22
Therefore, if there is no coupon
firms have no liquidity demand past date 1
before date 2 (so
risk (0 = 1) or if there is no signal about the realization of
that there is a coupon risk, but no price risk at date 1), short-term and longterm bonds will be perfect substitutes and so q = Q. Suppose instead that the
Now the price at which long-term bonds
realization of 6 is learned at date 1
.
21 The analysis in this section is valid for any concave private benefit function.
22
This example is meant to illustrate a situation in which most of the liquidity
to
be employed
in
is expected
the short run with long-term liquidity needs expected to be less pressing.
20
E
can be disposed of at date 1, namely 6, will vary. The coupon risk in this case
induces a price risk.
Let £ and L denote the number of short-term and long-term bonds purchased
by the corporate sector (in equilibrium, — I if q > 1 and L = L if
at date
(.
Q>
1.)
The corporate
sector solves
max
{vMM
(V(*)Y
Eq by(x)
s.t.
Eo[x-I-y{x)-(q-l)e-(Q-l)L]>0,
and
y(x)
Let
<x + £ + 6L
for all x.
again denote the shadow cost of the investors' break -even constraint and
/*
= b — denote the optimal unconstrained reinvestment level. Because
=
I
^and L = L in equilibrium, equilibrium prices are characterized by:
let
y*
j.L
r
q-1
y*-e-8L
b-(x + l+0L)
-r\i
Jo
r
Q-i
=f
z'{6)
<
0.
g{x)dx
h(6)d0,
(28)
b-{x + e + 6L)
g(x)dx
h(e)d6.
(29)
Jo
fi
integrals in (28)
and
(29)
by
It is easily verified
z(6).
that
Consequently,
Q-1=E
The
y*-l-6L
9
Jo
Denote the inside
-1
M
Jo
[Bz(B)\
<E
\z{B)\
[9]
price differential between short-
more general theme, namely that the
=E
[z{6)]
= q-l.
and long-term bonds here
(30)
reflects
a
approach to asset pricing implies
a skewness in risk tolerance. What matters is the average coupon delivered in
states of pressing liquidity need. The term premium stems from the fact that a
short-term bond delivers 1 unit for sure in such states, while the date-1 price of
the long-term bond is negatively correlated with the marginal liquidity service
m(-). The difference q — Q measures the cost of hedging against the date-1
coupon/price risk on long-term bonds.
Non
neutrality of price risk.
service (namely the
same
tainty about inflation or
liquidity
Long-term bonds would provide a better liquidity
if there were no uncerThis
no information about 6 arrived at date 1.
service as short-term bonds)
if
21
contrasts with an Arrow-Debreu economy, in which early arrival of information
endowment economy,
a production economy because of improved
decision making. Our model features a logic similar to Hirshleifer's (1971) idea
that early information arrival may make agents worse off.
Our model differs
from Hirshleifer's, in that in his model information arrives before entrepreneurs
never
is
harmful. Such information has no impact in an
and may generate
social gains in
and investors sign a contract. In our model
to bringing in funds at date
1
Put
tion leakage problematic.
it is
the investors' inability to commit
that constrains contracting and makes informadifferently, investors
cannot
offer entrepreneurs
insurance against variations in the price of long-term bonds.
long-term bonds
is
the price of
If
negatively correlated with the firm's liquidity needs, as
is
the case here, long-term bonds become inferior liquidity buffers to short-term
bonds.
Neutrality of pure price risk. Let us follow section 4
the date-1 state accrues between date
that
is,
at each subdate n, a signal
and date
and assume that news about
1,
an accrues that
at subdates
n
=
l,...,N;
informative about the
is
date-1 income and/or the coupon on the long-term bond. Then, the prices of
and long-term bonds adjust from their date-0 values q and Q to q(crn )
and Q{an ). This price risk, however, has no impact on the date-0 prices q and Q
which remain given by (28) and (29), since the corporate sector in equilibrium
short-
does not reshuffle
its
portfolio of liquid assets (this
is
the point
made
earlier
that capital gains and losses on financial assets have offsetting effects.) In other
words, price risk has no impact on the slope of the date-0 yield curve.
other hand, news
The
On
the
affects the slope of the yield curve at subdates.
may
upward sloping even in the absence of a coupon risk. In
risk, q = Q. Let i\ and i-i denote the yields on short
and long bonds at date 0. These yields are negative in our model because the
yield curve
be
the absence of any coupon
consumers' rate of time preference
q
is
normalized to zero.
1
-n
Q -l+h-
We
have
1
(l+i 2
f
and so
ii
=
<7
This
trivial
~
K h = —m
VQ
example makes the point that
a necessary condition
for
1.
riskiness of long-term
the existence of a term premium.
bonds
The
is
not
yield curve
can be upward-sloping, as here, simply because the corporate sector has no
liquidity demand at date 2. This suggests that an upward slope is associated
with relatively more pressing short-term liquidity needs, perhaps because the
firm has less flexibility to adjust plans in the short term.
22
Other shapes of the yield curve. Suppose the income shock x and reinvestment
and the private benefit accrues at date 3. Investments and financing still occur at date 0. In this temporal extension of the
model, date 1 is just a "dummy date," at which nothing happens. Suppose that
the government still issues short-term bonds (maturing at date 1) and long-term
bonds (maturing at dates 2 and 3). Short-term bonds offer no liquidity service
and so q = 1 Hence the short rate (equal to 0) exceeds the long rates, and we
decision y take place at date 2,
.
obtain an inverted yield curve.
This example makes the simple point that
pect to face liquidity needs in the short run,
on short-term securities, and so they
the corporate sector does not ex-
if
it
does not pay a liquidity premium
more than long-term securities.
will yield
Coordination failures in the creation and
6
uti-
lization of liquidity
The
analysis so far has
assumed that the corporate sector makes
of liquidity, but has no control over the aggregate supply.
assumption
for assets
such as Treasuries whose
This
is
efficient
supply Lk and
(i)
use
a reasonable
(ii)
state-
contingent payoffs 9k(u>) can be considered exogenous. We now show by means
of simple examples that the corporate sector's date-1 policy can affect aggregate
liquidity in
ways that are relevant
for policy
making.
In the first example, the
aggregate supply of liquidity can be directly influenced by the corporate sector.
In the second example, the effect
decisions. In
is
indirect as asset prices
both cases, coordination
may
failures
we continue to assume that the corporate
sector
depend on corporate
occur despite the fact that
makes optimal use of
its
liquid
assets.
Throughout
this section, there are
no bonds or other forms of external
liq-
uidity.
6.1
Precarious employment relationships hurt long-term
savings and equity investments
In this section only, assume that consumers, whose mass
some "subsistence level" (food, education, housing, etc.)
date. That is, we replace the utility
Co
+ Ci + c2
from consumption flow (co,Ci,C2) by the
co
utility
+ u(ci) + c2
where
23
,
is
1,
must consume
at the intermediate
1
11(d)
=
if cj
J
{
>
Cj
>
(31)
otherwise.
C
Consumers thus care about the consumption path as
The
As
representative firm's investment cost at date
well as its level.
is
normalized at I
=
1.
assume that consumers have enough of the nonstorable good at date
to help finance the initial investment /. The firm's income at date-1 has two
possible values: high income xh with probability fff, and low income xl with
probability fi = 1 — fn where
before,
,
xH > xL
Continuation at date
A
(31)
and
will
Continuation yields a private benefit
As
well,
1
(33)
.
imply that consumers do not value liquidity at date
(33)
provided that they know that they
(pledgeable) income
(32)
.
one worker /
worker must be paid an efficiency wage w,
w>c
1
c1
requires, in addition to entrepreneurs,
1
consumer per unit of investment.
where 23
Assumptions
=
X. Assume
have a job at date
B
1.
to the entrepreneur,
B+X > w
and a
so that continuation
is
verifiable
optimal. 24
assume that
X - w < 0,
(34)
+ X - w > 0,
(35)
+ fH (xH+X-w)>0.
(36)
xL
and
-I
With these parameter
and the maturity
liquidity
•
A
restrictions there
a feedback between aggregate
long-maturity, high- liquidity, high-employment equilibrium.
Suppose,
first,
that
all
firms continue at date
consumers then have a job at date
23
is
of savings such that multiple equilibria can arise.
1
and
We
1
receive a
in
both states of nature. All
wage w.
By
(31)
and
(33)
can invoke the standard efficiency wage model: The worker receives some private
none when working. Suppose for simplicity that the work/shirk
decision is perfectly verified ex post, that the worker is protected by limited liability, and that
shirking has disastrous consequences for production and must be prevented. Then firms must
pay at least w to their workers.
2
24
In the notation of section 2, y =
or w, B = bw 2 - [u> /2). In section 2, we assumed
= 0. Here X is assumed positive.
benefit
w when shirking and
X
24
,
they have no
demand
for liquidity
and are thus
willing to defer all
payments on
their date-0 investment to date 2.
Since the corporate sector need not meet any short-term payment obligations,
from
X
>w
always has enough liquidity to pay the date-1 wage w, as Xl +
that
the
firms
can
repay
date-0
Conditions
and
then
imply
(35).
(35)
(36)
it
investors out of
X since
—I + [Jh^h +
Il^l]
+ X — w >0.
In this (efficient)
equilibrium the corporate sector issues long-term claims and does not lay off
workers.
A
short-maturity, low-liquidity, low-employment equilibrium.
Suppose instead that firms are unable to continue in the bad state of nature.
Consumers become unemployed and because they have to to consume c x they
Therefore, they
will insist on receiving at least c x in this bad state of nature.
want to hold short-term claims on firms. Condition (32) guarantees that this is
indeed feasible, while (34) and (36) imply that given that workers are paid Cj
firms no longer have any cash to finance the reinvestment. Firms continue only
in the good state of nature, and equation (36) assures that such a plan can be
•
,
financed at date
This
in
0.
(inefficient)
25
equilibrium exhibits a coordination failure.
It illustrates
a stark manner that the liquidity available to the corporate sector depends on
the
liability side as well as
the asset
side.
In this respect,
it is
interesting to note
that a recent speech by the French finance minister 26 called for "well-oriented
meaning a switch by households from short-term market investments
towards long-term, equity investments that will benefit the productive sector. 27
savings,"
A
First, the government can
unemployment insurance. Unemployment
here eliminates the consumers' demand for liquidity and
couple of additional remarks can be made.
create aggregate liquidity by offering
insurance (above Cj)
restores the efficient equilibrium. Second, the multiplicity of equilibria could not
an Arrow-Debreu economy, because in a perfect capital market, firms
would be able to pledge their entire date-2 income which would be enough to
pay workers at date 1 in both states.
arise in
One might wonder whether this coordination failure could be avoided if workers invested
own firm at date and signed a contract with their employer specifying that
solely in their
will be withdrawn at date 1 as long as they are not laid off. This arrangement is
not robust to minor perturbations of the model such as job mobility or idiosyncratic liquidity
shocks. Suppose for example that with a small probability each firm receives no income at all
at date 1 Then the workers, who want to secure £j must diversify their portfolio (have claims
at least worth Cj in other firms), and the coordination failure may still occur. [Technically,
one must assume that xl exceeds c.jslightly so as to offset the absence of income in this small
number of firms and allow all consumers to "survive" in the bad state of nature.]
Jean Arthuis January 14, 1997 speech at the parliamentary meetings on savings.
The speech discussed several ways of encouraging equity investments, such as the creation
of pension funds and the reform of the tax system (equity investments in France are taxed at
the personal income rate; this implies an overall tax rate of 61.7% for the highest tax bracket.
In contrast, money market funds are taxed in a lump-sum fashion at a rate not exceeding
20%).
no cash
,
.
1
25
6.2
In
Liquidity creation through price support policies: the
example of commercial real estate.
many
countries the recessions of the late 80s
and early 90s have
left
nancial institutions burdened with depreciated commercial real estate.
the
fi-
While
banks, badly in need of liquidity, would have liked to divest their real estate
holdings, they realized collectively that
dumping
real estate assets
on the mar-
ket simultaneously would have a disastrous impact on prices in a state of low
demand
commercial real estate. In some countries (e.g., in France), cartelon the disposition of real estate prevented prices from falling
28
further.
To an industrial organization observer, this behavior has all the attributes of a price-fixing case. This section argues that there is more to it than
just collusion and that price stabilization may actually have helped to improve
economic efficiency.
To illustrate the point, suppose that, as in section 6.1, continuation at date
1 requires paying for an input, but this time, let the input be commercial real
estate rather than labor. In case of continuation, one unit of commercial real
estate is needed.
with
The date-0 investment yields date-1 income x^ =
probability fi and Xh >
with probability fjf = 1 — fz,- As in section 2,
reinvestment yields a private benefit B at date 2, but no pledgeable income
(X = in the notation of section 6.1). The reinvestment cost is p + e, where
p > and e is the price of commercial real estate. Letting ei and e# denote the
commercial real estate prices in the bad and good states, the overall liquidity
need is then
for
like restraints
p
+ ei
in
the bad state (probability ji)
or
p
+ en — xh <
in
the good state (probability ///)
Commercial real estate construction is part of the initial invesment. Each firm
buys one unit of real estate per unit of investment. Firms invest this amount in
because they want to stand ready to produce
commercial real estate at date
at least in the good state.
On the consumer side, we return to our basic paradigm in which preferences
are linear:
Co
+ C\ + C2-
Divested real estate
is
costlessly converted into residential real estate
on a
To make our main point in the starkest way, suppose that
(residual) demand at price v\ the absorption capacity of the
one-to-one basis, say.
there
is
a fixed
residential real estate
market
is z.
real estate is converted, the price
more than z is converted, there
drops to 0. Assume that
is
than z units of commercial
market is v; if
excess supply and the price on that market
That
on the
is,
if
less
residential real estate
Such cartels are sometimes organized by the Central Bank.
26
v>(l-z)(p + v).
(37)
Again, there are two possible equilibria:
• Low-price, low-production equilibrium:
Suppose that
in
the bad state the corporate sector
real estate onto the residential market.
The
no other liquid asset besides
real estate all firms
the commercial real estate
now
is
free,
dumps
all its
commercial
and since there is
are liquidated. Even though
price drops to 0,
firms are unable to continue.
• High-price, high-production equilibrium:
Suppose instead that only a fraction z of the assets are liquidated, where
v=
(1
- z)(p + v)
Conditions (37) and (38) imply that z < z, and so the market price of real
is v. The corporate sector thus has total available liquidity per unit of
estate
investment equal to v, and can finance the shortfall p + v on a fraction
1
—
z of
its assets.
In this equilibrium, consumers pay higher date-1 prices for residential real
estate
and the
against
restraint
on
sales provides insurance to the corporate sector
credit rationing at date
full
l.
29
Such insurance
raises ex ante social
surplus.
Concluding remarks
7
For a long time, corporate finance has been treated as an appendix to asset
pricing theory, with CAPM frequently used as the basic model for normative
analyses of investment and financing decisions. While standard textbooks still
reflect this tradition,
fluence the
the modern agency-theoretic literature is starting to infinance is taught. This paper takes the next logical
way corporate
is to suggest that if financing and investment decisions reflect agency
problems
as seems to be widely accepted
then it is likely that modern corporate finance will require adjustments in asset pricing theories, too.
step, which
—
Our paper
—
is
a very preliminary
in
effort to
analyze this reverse influence of
We have employed a standard agency
model
which part of the returns from a firm's investment cannot be pledged to
corporate finance on asset pricing.
outsiders, raising a
29
This example
demand
for
long-term financing, that
is,
for liquidity.
We
and Moore's 1997 analysis of the dual role of
and an input into production. Our emphasis differs both in
the ingredients (our treatment relies on the existence of aggregate shocks while theirs does
not) and in the emphasis (they stress the possibility of business cycles while we emphasize
market power and liquidity creation through price support policies.).
is
in the spirit of Kiyotaki
certain assets as a store of value
27
have also assumed that individuals cannot pledge any of their future income, so
human capital is impossible. As a result, the economy is
typically capital constrained, implying that collateralizable assets are in short
that borrowing against
supply.
Such assets
will
command a premium, which
is
determined by the
covariation of the asset's return with the marginal value of liquidity in different
Risk neutral firms are willing to pay a premium on assets that help
This is a form of risk aversion, but unlike
in models based on consumer risk aversion, return variation within states that
experience no liquidity shortage is inconsequential for prices. Thus, liquidity
states.
them
in states of liquidity shortage.
premia have a built-in skew.
One consequence of this skew
we
is
that price volatility tends to be higher in
Another consequence
sell at a discount
relative to short-term bonds as we showed in Section 4. This may be one reason
why the yield curve most of the time is upward sloping, a feature that does not
readily come out of a complete market model.
The price dynamics in our model satisfy standard Euler conditions
in
particular, prices follow a martingale as long as there is no readjustment in the
corporate sector's coordinated investment plan. It is an interesting possibility
that marginal rates of substitution for the corporate sector may be quite different, and perhaps more volatile in the short run, than the marginal rates of
substitution of a representative consumer. This could help to resolve some of the
empirical difficulties experienced with consumption-based asset pricing models,
which appear to feature too little variation in MRSs.
Our model is quite special in that asset prices are entirely driven by a corporate demand for liquidity; consumers hold no bonds or other assets that sell at
a premium. It has been suggested to us that once the model is changed so that
consumers also have a liquidity demand, MRSs of consumers and firms will be
equalized, and we are back to the old problem with excess asset price volatility.
However, if consumers participate selectively in asset markets, then the MRSs
of the relevant sub-population may have high volatility and yet be hard to detect. In this case, the equality between consumer and producer MRSs can be
exploited in the reverse: by evaluating corporate MRSs, we can infer what the
MRS of the representative consumer in the sub-population is. This may be a
useful empirical strategy if firm data are more readily available and easier to
analyze than consumer data.
states of liquidity shortage, as
is
illustrated in Section
that long-term bonds, because of a higher price
risk,
3.
tend to
—
Finally,
we note that
violations of the martingale condition, as illustrated
by the end-of-period drop in the liquidity premium, may help to explain the
well-known paradox that prices of long-term bonds tend to move up rather
than down, following a period in which the yield spread (long/short) is exceptionally high.
This finding
is
difficult to reconcile
very
with the standard
expectations theory (Campbell, 1995), but could perhaps be accounted for in
a theory where liquidity demand shifts between short and long instruments in
response to changes
in liquidity needs.
28
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Management Techniques".
Appendix: Treatment of the productive sector as an aggregate
Suppose that there are n firms, i = l,...,n, each run by an entrepreneur,
Firm i starts with initial wealth A{ and invests Ii7 so that the net outlay
for investors is Ni(Ii) = Ii — A{. At date 1, in state w, firm i takes a decision
di(u) in a subset Di(u,L™(uj)) of the technologically feasible set Di, where
is the net liquidity available to firm i in state u).
L"(u>) >
means
Lf(u>) ^
that firm i uses liquidity, and L™(u>) <
means that firm i supplies liquidity
in state u. The decision di(u>) generates an expected pledgeable income from
productive assets Ti(Ii, uj,di(ui)) and a total expected income from productive
say.
Let
assets Ri(Ii,u),di(u>)).
Bi(Ii,u),di(u}))
=
ili(/i,w di(w))-7- i (/i ,w,d i (w))
!
i that must go to entrepreneur i. Endenote the
may, however, be paid more than B{. Let ij(w) >
expected transfer on top of the nonpledgeable income Bi.
(^i(-) could without loss of generality be chosen equal to
in the example.)
So entrepreneur i
denote the nonpledgeable income of firm
trepreneur
i
obtains, in state
ti(u))
u>,
Bi(Ii,u>,di(ui))
from noncorporate
+ ti(ui).
If
firm
i
withdraws gross liquidity
assets, then the net liquidity available to the firm
is
L?(u)=Li{u)-ti(u>).
The economically
feasible set for firm
wants to enter negative
Di{u,Li{u)
NPV
- U{cj)) =
i
reflects the fact that
investments at date
{di
G Di
|
no investor
1:
n(Ii,u),di(w))
-
ti(w)
+ Li(uj) >
0}
Let
D{u,L n (u)) =
|
xSUA(w,£?M)
I
J>?M ^ Ln ^
\
denote the product decision set, with generic element d(u)). Consider a coordinated plan with weights {wi} i=1
n defined as the solution to the following
Program I
max
{l.,L.,d{)A)}
<
Eq
Y^Wi
[Bi{Ii,u),di{u)))
+ U{u) -
Ai]
I
s.t.
Eo ^ri(Ii,u,di(u)))
Z^lIi-AA+Y^fa-VLk
and
32
.
Letting
fi
respect to
denote the multiplier of the break-even constraint, optimization with
L k together with the equilibrium condition L k — L k yields
,
qk
-\ =Eo[m(u})6k (<j)],
where
m(w)
= Tl
f
max
Ei^gilijIEiii)
]
d€D(u,L)
The
as in equation (9).
L=Y. k
first-order condition
e k {u>)L k -*E
with respect to ti(u)
t i (u,)
i
is:
is:
either
or
Now assume
(this is
m(u>)
>
=
ti(u)
>
and
and
W{
Wi
< /x [1 + m(u)]
= [1 + m(u>)]
fi
that there exists at least one state of nature with excess liquidity
the case
That
0.
U(ui)
in all
is,
Then
our examples).
pledgeable income
states of liquidity shortage.
is
The marginal
for all
ti(u>)
i,
=
whenever
never distributed to entrepreneurs in
liquidity service
can then be rewritten
as:
d
l
max
\
dL
<
J=Lk
d£D(u>,L) [
Thus the equilibrium
is
pi
L=-£,e k (u>)L k
as described in the text, with
B = ^^iBi,
and
r
=
Remark: In Program I, we did not consider individual rationality constraints for
the entrepreneurs because such constraints only affect the set of feasible weights
Wi.
Let us
now show
that a decentralized equilibrium of a market for state-
contingent liquidity can be construed as an optimal coordinated plan. Suppose
that the firms at date
contract on their date-1, state-contingent liquidity in a
perfect market. Let m(u>) denote the net cost of one unit of liquidity in state
w.
That
£b[(l
is,
in
order to receive Li(u>) in state
+ m(u)))Li(w)]
at date 0.
Firm
i
u>,
for all
therefore solves:
{Ii,Li(),di[),ti{)}
S.t.
33
o>,
firm
i
must pay
MIT LIBRARIES
3 9080 01444 0777
Eo[ri{Iuu,di{u))
- ti(w)] >Ii-Ai + EQ [m(uj)Li(uj)\
and
di{u))
£Di(w,Li(u)-ti(u))).
Letting l/wi denote the shadow price of the budget constraint, this program,
under our concavity assumptions, is equivalent to
Program II
Ui.-M'), <*((•), ii(-)}
+r
u, di
(Ii,
t
(<j))
-U
(u)
-
U-m (w) Li(u)]}
s.t.
£ Di
di(w)
—
(w, Li(u)
for all w.
ti(w))
Given the existence of a state Wo suc h that m(wo)
and Li(u) implies that u^ < 1 for all i, and ti{bj) =
Summing
over the individual programs,
we
=
0,
optimizing over
for all
w
if
Wi
<
ti(u>)
1.
see that the market equilibrium
solves
max£b Y^ Wi
[Bi
+ ti] + Y, h - r - "»(w)
t\
2L
<
s.t.
di{u>)
FVom our previous
£
Di(u),Li(ui)
— tj(cj))
for all u.
characterization,
Y( w i-
l ) t i{ UJ )
=°
for all w,
t
which implies that the constraints di S Di can be replaced by d £ D, as defined
Therefore the market equilibrium solves Program I for some weights
earlier.
Remarks:
1.
2.
This equivalence result would not hold, necessarily, if the set of liquid
instruments did not span the state-contingent claim space. In this sense,
we are assuming that the state-contingent liquidity space is complete.
The
weights {wi} are endogenous, so one cannot conduct comparative
and we do not. However, the
statics exercises with this representation,
analysis of price dynamics in Section 4
balancing of portfolios.
34
is
valid,
because
it
entails
no
re-
x
—Date
x
—Date
• Set
up cost
/
and choice
•
disburse /
+
qL.
x
Random income
Total income
of liquidity L. Investors
—Date
1
is
x
+
L.
•
x
+L—
Figure
• Private benefit
2
by — y /2 for the
entrepreneur.
x.
Reinvestment y(x).
Payout to investors
•
2
y(x).
1
Marginal liquidity service m(x)
m Ll (x)
V
m L3 (x)
First period
income x
Figure
k
8 7
£
[
:
I
2:
L 3 > L* > L 2 > L
x
Date Due
Lib-26-67